Quantum Monte Carlo in Condensed Matter Physics:

Past, Present, and Future

0. Overview

1. Past: Algorithms and Algorithm Development

• World-Line Methods

• Auxiliary Field Methods

2. Present: Results- Where are we?

• World-Line Methods

• Auxiliary Field Methods

3. Future

• Algorithmic Challenges

• Computational Physics Education

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Mohit Nandini Thereza George DavidRanderia Trivedi Paiva Batrouni Cone

Marcos Simone Chris Khan EliasRigol Chiesa Varney Mahmud Assmann

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0: OVERVIEW: Progress in Superconductivity: Tc versus Time

Rapid acceleration in field(Note change in scales!)

Many new materialsHeavy fermionsCuprates (high Tc)Iron-Pnictides · · ·

New mechanismsElectron-phonon mediated →Spin mediatedCharge inhomogeneities

200

150

100

50

40

30

20

10

1900 1940 1980

1985 1990 1995 2000 2005 2010

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OVERVIEW: Progress in World Line Quantum Monte Carlo

Worm

1980 1990 2000 2010

Year

SF−MI

WLQMCSupersolids

2D Bose−Hub

SF

HeliumSupersolids

Hubbard

1D Holstein

Supersolids

AFLRO2D Heis

Ring Exch

2D Bose−Hub

Spins

SpinLiquids

SpinsDimerized

InteractionsMultispin

DeconfinedQuantum

Criticality

EntanglementEntropy

Bose Liquid

Tri Bose−Hub

Disordered

Disordered

1D Fermion

Helium 1D Bose−Hub

2D Bose−Hub

Critical pt

SSE

Loop

Same time frame as ‘modern’ SC era:Algorithm development:

Initial Formulations: 1982-84Advances SSE, Loop, Worm: 1989-96

Rapidly Accelerating ApplicationsCritical Points (eg SF-MI) to 0.1%Settle Bose Glass controversyNew Paradigms for Phase Transitions

Except for fermions in d > 1 !!

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OVERVIEW: Progress in Auxiliary Field Quantum Monte Carlo

1980 1990 2000 2010

Year

HubBilayer

3D HubEntropy

LaCoO3

2D Hubn(k)

LGT

AFQMC(x,t)

AFQMC(q,t)

BaFeCoAsCeIrIn5

SrRuO4LaSrCuO

MnO

α δPu

AFLRO

AF2D Hub

AFQMCDFT

Merger

AFQMC(.,t)

2D Hub

2D Hubd−wave

2D Hubd−wave more

PAM

Algorithm development

102 → 103 sites/orbitals

Merger with DFT

Apps: materials on SC timeline!

2D sign problem much better than WLQMC

Lattice Gauge Theory ‘Cousin’

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Providing Context: The Hubbard Hamiltonian

H = −tX

〈ij〉σ

(c†iσcjσ + c†jσciσ) + UX

i

ni↑ni↓ − µX

iσ

niσ

Operators c†iσ (ciσ) create (destroy) an electron of spin σ on site i.

Includes electron kinetic energy (t) and interaction energy (U).

Ut

U large favors local moments: Single ↑ or ↓ per site.What is optimal spin arrangement?

Hopping of neighboring parallel spins forbidden by Pauli.

Antiparallel arrangement lower in second order perturbation theory.

x t t

∆E(2) = 0 ∆E(2) = −t2/U = −J

Antiferromagnetism is accompanied by insulating behavior.

Transition Metal Oxides: AF insulator (Hubbard’s motivation).

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Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finite

0-6

Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finite

0-7

Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finite

0-8

Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finite

0-9

Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finite

0-10

Mott Insulator

U/t large and 〈n〉 = 1.

All sites occupied by exactly one e−.

Hopping causes double occupancy, costs U .

Chemical potential µ

∗ Cost to add particle

∗ Jumps at ρ = 1

-8 -4 0 4 8µ

0.5

0.75

1

1.25

1.5

ρ

"Mott Plateau"

U = 18 tT = 0.5 tN = 8x8

κ = dρ/dµ = 0

κ finite

κ finiteSummary: Hubbard model

• ρ = 1: AF-insulator

• ρ 6= 1: d-wave SC ???

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1. PAST: Path Integral Quantum Monte Carlo Simulations

Partition Function

Z = Tr e−βH

Inverse Temperature discretized: β = L∆τ

Z = Tr [e−∆τH ]L = Tr [e−∆τHe−∆τHe−∆τH · · · e−∆τH ]

Trotter approximation H = Ha + Hb + · · ·Hm

e−∆τH ≈ e−∆τHae−∆τHb · · · e−∆τHm

Individual e−∆τHx calculable.

Extrapolate ∆τ → 0 (but see later · · · )

World line and auxiliary field methods differ in how individual e−∆τHx treated.

Here: Focus on lattice models. (Continuum, eg Helium, Ceperley)

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World-Line Quantum Monte Carlo Simulations

e−∆τHx calculated by spatial separation:

{Hx} contain independent small spatial clusters, e.g. 1D Hubbard

H = −tX

iσ

(c†iσci+1σ + c†i+1σciσ) + UX

i

ni↑ni↓ − µX

iσ

niσ

Independent

1 2 3 4 5 6 7

Ha (1,2)

Hb

(2,3) (4,5) (6,7)

(3,4) (5,6)

2 site clusters

Insert complete sets of occupation number states

Z =X

nl

〈n0|e−∆τHa |n1〉〈n1|e

−∆τHb |n2〉〈n2|e−∆τHa |n3〉〈n3|e

−∆τHb |n4〉

. . .〈n2L−2|e−∆τHa |n2L−1〉〈n2L−1|e

−∆τHb |n0〉

〈nl|e−∆τHx |nl+1〉 breaks into product of independent cluster problems.

Sign problem: Are 〈nl|e−∆τHx |nl+1〉 positive?

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State of system represented by occupation number paths ni(τ)

Paths sampled stochastically (locally distort world lines).

Weight: product of matrix elements.

1 2 3 4 5 6 7x

τ

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7x

τ

1

2

3

4

5

6

7

8

(a) (b)

Zero Winding Non-Zero winding (SF density nonzero)World line topology linked to underlying physics (Ceperley, Pollock)

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Key features of (original) World Line QMC:

• Linear scaling in particle number (system size).

• (Very) Long autocorrelation times.

• Cannot measure Green’s function Gσij = 〈c†iσcjσ〉 (would break world lines).

• Sign problem for fermions and frustrated quantum spins.

Advances:• Continuous time algorithms eliminate β discretization. No Trotter errors.

• Loop and worm algorithms improve sampling.Autocorrelation times decrease by 3-4 orders of magnitude.Sample nonzero winding sectors.

• Extend measurements.Superfluid density ρs.Greens function (‘worms’ allow ‘broken’ world lines); n(k).

Bottom line:

• Can simulate 104 − 106 quantum bosons/spins.

• Very high precision on critical points, exponents.

• Address very subtle issues in nature of phase transitions.

• Sign problem remains for fermions.

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Auxiliary Field Quantum Monte Carlo

Separate H = K + P into kinetic and interaction pieces.

• (Discrete) Hubbard-Stratonovich Fields decouple interaction:

e−∆τUni↑ni↓ =1

2e−

U∆τ

2(ni↑+ni↓)

X

Siτ

e∆τλSiτ (ni↑−ni↓) = e−∆τPi(τ)

where cosh(∆τλ) = eU∆τ

2 .

• Quadratic Form in fermion operators: Do trace analytically

Z =X

{Siτ}

Tr [e−∆τKe−∆τP(1)e−∆τKe−∆τP(2)e−∆τK . . . e−∆τP(L)]

=X

{Siτ}

detM↑({Siτ})detM↓({Siτ})

dim(Mσ) is the number of spatial sites.

• Sample HS field stochastically.

Si0τ0→ −Si0τ0

detMσ({Siτ}) → detMσ({Siτ}′)

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Key features of (original) Determinant QMC:

• Algorithm is order N3L. (Computation of determinants.)

• N ∼ 102 lattice sites (fermions).

L = β/∆τ ∼ 200 (to reach low temperatures).

Ground state projection methods (Sorella, S-W. Zhang)

• Sign Problem

At low temperature detMσ can go negative.

However, typically occurs at T ∼ t/4.

Order of magnitude lower T than WLQMC!

T cold enough to characterize short range spin/charge correlations.

Special symmetry cases (U > 0, ρ = 1; or any U < 0, all ρ): sign detM↑, detM↓ same.

Constrained Path Approaches (S-W. Zhang)

Advances:

• N ∼ 102 → N ∼ 103 fermions (collaborate with CS researchers: D’Azevedo; Bai)

• Integration with Electronic Structure Codes via DMFT

• Better scaling in β (Khatami)

• Continuous time solvers for DMFT.

Treat strongly correlated material rather than models.

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2. Present WLQMC

2D Boson Hubbard model ground state phase diagram (bosons- no sign problem)

t U

U

3U

t

Weakly interacting bosons (U small): Superfluid (SF) phase.

MI

SF

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2. Present WLQMC

Boson Hubbard model ground state phase diagram.

t

UX

Strongly interacting bosons ( (U large, ρ = 1): Mott Insulator (MI) phase.

Locate position of SF-MI (U/t)crit to a small fraction of a percent.

Capogrosso-Sansone etal.

MI

SF

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Disordered Boson Hubbard model ground state phase diagram.

Disordered site energies (∆ 6= 0) induce additional Bose Glass phase.

Both order parameters (Superfluid density and Mott gap) zero.

Precision determination of ∆crit.

Bose glass phase always intervenes between MI and SF.Prokof’ev etal.

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“Deconfined Quantum Criticality” (DQC)

Transition between 2 competing phases doesn’t follow traditional Landau description.

-20 0 20 40

L1/ν

(q-qc)/q

c

0.5

1

1.5

2

2.5

3

M2 L(1

+η s) , D

2 L(1+

η d)

L = 24L = 32L = 48L = 64

-20 -10 0 100

0.5

1

1.5

2

2.5

M2 L(1

+η s) , D

2 L(1+

η d)

L = 24L = 32L = 48L = 64

Spin

Dimer

Spin

Dimer

J-Q2

J-Q3

Interest: Could DQC underlie physics of high Tc and heavy fermion superconductors?!

Competition between AF and pairing order parameters

Unusual behavior observed in vicinity of transition.

Hamiltonian with competing AF (spin)

and valence bond (dimer) phases.

Scaling collapse demonstrates DQC

in appropriate spin models.

Sandvik.

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2. Present AFQMC

Recent Progress in Lattice Gauge Theory

(Auxiliary Field Quantum Monte Carlo Cousin)

Last five years: computation of parameters of Standard Model

Cabibbo, Kobayashi, Maskawa (CKM) matrix elements.

Examine effects of QCD on weak interactions.

Close methodological connections to AFQMC.

LGT: Quarks and gluons on a lattice.

CM: Electrons and phonons (or Hubbard-Stratonovich field) on a lattice.

LGT size now at 643 x 192 versus 322 x 192 for Condensed Matter AFQMC.

Reason is that LGT has linear scaling algorithm!

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Decay of D meson (charmed + light quark)

to K meson (strange + light quark), lepton, and neutrino

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

q2/m

Ds*

2

0

0.5

1

1.5

2

2.5

f +(q

2)

q2

max/m

Ds*

2

lattice QCD [Fermilab/MILC, hep-ph/0408306]experiment [Belle, hep-ex/0510003]experiment [BaBar, 0704.0020 [hep-ex]]experiment [CLEO-c, 0712.0998 [hep-ex]]experiment [CLEO-c, 0810.3878 [hep-ex]]

D → Klν

• Overall normalization measures CKM matrix elements.

• Functional dependence on q2 (outgoing lepton momentum)

matches between LGT (2004) and experiment (2005-2008).

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Lattice (Hubbard) Models in Condensed Matter

Increased DQMC lattice size: resolution of Fermi surface (occupied/empty k)

U = 4 Fermi function:

ρ = 0.2 β = 8

-π -π/2 0 π/2 π-π

-π/2

0

π/2

πρ = 0.4 β = 8

-π/2 0 π/2 π

ρ = 0.6 β = 6

-π/2 0 π/2 π

ρ = 0.8 β = 4

-π/2 0 π/2 π 0

0.2

0.4

0.6

0.8

1

n(k)

ρ = 1.0 β = 8

-π/2 0 π/2 π

U = 4 Gradient of Fermi function:

-π -π/2 0 π/2 π-π

-π/2

0

π/2

π

-π/2 0 π/2 π -π/2 0 π/2 π -π/2 0 π/2 π 0

0.5

1

1.5

∇ n

(k)

-π/2 0 π/2 π

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Antiferromagnetic spin correlations form at low temperature.

βt = 12, 20, 32 T/t = 0.083, 0.050, 0.033.

-0.15

-0.1

-0.05

0

0.05

0.1

(0,0) (10,0) (10,10) (0,0)

C(l

x,ly

)

20 x 20 U = 2.00

(0,0) (10,0)

(10,10)

(a)

β = 32β = 20β = 12

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Dynamic Cluster Quantum Monte Carlo: superconductivity in the 2D Hubbard model

What is the interplay between stripes and pairing?

Nonzero V0 (charge inhomogeneity scale) initially increases Tc.

Dynamic Cluster Approximation: 100 site momentum space clusters.

Can do 100 site momentum space clusters. Maier etal; other DCA: Jarrell etal.

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Combining Density Functional Theory and AFQMC

Hkin + Vion

(e.g. in LDA basis)

Wannier projectionHkin + Vion → H0(k)

VCoul.(r − r′) → Vij

AIM: Γir(ω, ν, ν′)

Parquet Eq.:with Vij + Γir − Vii

Eq. of MotionDyson Σ(k, iω); G(k, iωn)

Gloc. =∑

k

1

iωn − H0(k)

Glo

c.

=∑ k

G(k

,iω

n)

→new

AIM

Density Functional Theory: Compute band structure H0(k) for given material.

Dynamical Mean Field Theory: Computes additional self energy from strong correlations.

Renaissance of diagrammatic methods to refine/extend DMFT:

Dynamical Cluster Approximation

Dynamical Vertex Approximation (shown) · · ·

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DMFT Revealing Unexpected Physics in Multi-Orbital/Band Models

Two band Hubbard model with crystal field splitting (appropriate to LaCoO3)

“Spin Disproportionation”: At low temperature T sites equivalent (same spin).As T increases (!) order arises: alternation of high spin (magnetic) and low spin.

(nonmagnetic) sites at intermediate temperature.

10

20

30

χ (µ

B2 /eV/

atom

)

0 200 400 600 800Temperature (K)

0

0.05

0.1

na

0 5000

50

HS LS

Intermediate T has

Enhanced magnetic susceptibility

and different LS and HS occupations

∆cf = 3.40, (3.42) for red(black)

Kunes etal, arXiv:1103.2249

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“Linear Scaling Algorithms”

Many computational problems involve linear algebra

Dense Matrix Multiplication, Matrix inversion o(N3)

Can algorithms be formulated as o(N) ?

(Electronic Structure and Quantum Monte Carlo, for example)

General Argument that such algorithms might exist in principle:

“Nearsightedness principle” (Kohn 1996)

Influence of degrees of freedom in problem fall off sufficiently rapidly:

Partition problem into local spatial domains.

Issues in Practical Implementation:

How is domain size determined? Done for each separate problem?

How robust? Do small errors associated with partitioning blow up in “time”?

(eg as Molecular Dynamics or Monte Carlo simulation progresses)

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Example from Quantum Monte Carlo for fermions:

Z =

Z

Dx detM(x) ⇒

Z

Dx

Z

DΦ exph

− Φ(M(x)TM(x))−1Φi

x: Field coupled to fermions, eg phonons, gluons, Hubbard-Stratonovich.

M is a sparse matrix.

Φ update is trivial: Φ = MT R where P (R) ∝ expˆ

− RT R˜

x update requires computation of (MT M)−1Φ.

Do iteratively. Involves sparse matrix multiplication: → o(N) !

Works well in Lattice Gauge Theory.

Works poorly in simulations of the Hubbard Hamiltonian.

Number of iterations grows slowly with linear lattice size.

Grows very rapidly (even becoming unstable) in imaginary time.

Molecular Dynamics in LGT!

dx

dt= p

dp

dt=

d

dx

h

− Φ(M(x)TM(x))−1Φi

What happens at zero eigenvalues of M?

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Computational Physics Education

NSF, NAS emphasize the need for K-12 computing education:

“It’s one of the most vexing paradoxes facing the U.S. today, even if most people are notaware of it. American IT and software companies dominate the world market place andthe vast majority of colleges and universities have excellent computer science programs,yet at the K-12 level, computer science education is almost nonexistent.”

http://www.nsf.gov/news/news summ.jsp?cntn id=116059

Yet, computer programming classes remain absent from secondary schools,

or moving in the wrong direction.

Percentages of high schools offering:

2005 2010Introductory Programming Course 78% 65%AP Programming Course 40% 27%

Meanwhile, other countries have implemented a comprehensive (required)

secondary school computer science curriculum.

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Undergraduate and graduate level:

• 5-6 undergraduate programs in computational physics

• 25 minors/concentrations/tracks

But, in many cases, computational physics is not being emphasized in university physicscurriculum despite its increasing pervasiveness in research and in industry.

“we are teaching the same things we taught 50 years ago”.

“Report of the Joint AAPT-APS Task Force of Graduate Education in Physics”, June2006.

Rubin Landau/Steven Gottlieb

(editors of new series of textbooks incorporating computational physics)

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3. FutureAlgorithms

• Better treatments of dynamics e−βH → e−iHt (Rigol).

• Linear scaling algorithms for fermions in CM as exist already in HE (LGT).

• Solve sign problem.

Applications

• Continued study of exotic phase transitions for bosons/quantum spins.

• Entanglement entropy as a tool.

• Continued applications of DMFT+LDA method to strongly correlated materials.

Educating next generation of computational physicists.

[Richard Martin talk tomorrow]

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